Proof Using Complex Variables

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Proof Using Complex Variables

To show by means of phasor analysis that Eq. $\,$ (A.2) always has a solution, we can express each component sinusoid as

$\displaystyle A_i\cos(\omega t + \phi_i) =$ re $\displaystyle \left\{A_i e^{j(\omega t + \phi_i)}\right\}$

Equation (A.2) therefore becomes

$\begin{eqnarray*} \mbox{re}\left\{A e^{j(\omega t + \phi)}\right\} &=& \sum_{i=1... ...}\right\}\\ &=& \mbox{re}\left\{A e^{j(\omega t+\phi)}\right\}. \end{eqnarray*}$

Thus, equality holds when we define

$\displaystyle A e^{j\phi} \isdef \sum_{i=1}^N A_i e^{j\phi_i}. \protect$

(A.5)

Since $A e^{j\phi}$ is just the polar representation of a complex number, there is always some value of $A\geq 0$ and $\phi\in[-\pi,\pi)$ such that $A e^{j\phi}$ equals whatever complex number results on the right-hand side of Eq. $\,$ (A.5).

As is often the case, we see that the use of Euler's identity and complex analysis gives a simplified algebraic proof which replaces a proof based on trigonometric identities.

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Previous: Proof Using Trigonometry
Next: Phasor Analysis: Factoring a Complex Sinusoid into Phasor Times Carrier

written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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Introduction to Digital Filters
    Background Fundamentals
       A Sum of Sinusoids at the Same Frequency is Another Sinusoid at that Frequency
          Proof Using Complex Variables